Question

Classify the following pairs of lines as coincident, parallel or intersecting: $\left(i\right)2x+y-1=0and3x+2y+5=0\left(ii\right)x-y=0and3x-3y+5=0\left(iii\right)3x+2y-4=0and6x+4y-8=0.$

Answer 1

$\left(i\right)2x+y-1=0,3x+2y+5=0$

$\text{Writing equation in the form}y=mx+c$

$y=-2x+1,y=\frac{-3}{2}x-\frac{5}{2}$

$\Rightarrow m=-2,m^{\prime }=\frac{-3}{2}$

$m\ne m^{\prime },m_1{m}_2\ne -1$

$\Rightarrow \text{The lines are intersecting.}$

$\left(ii\right)x-y=0,3x-3y+5=0$

$\Rightarrow y=mx+c,3x-3y+5=0$

$y=x,y=x+\frac{5}{3}$

$\Rightarrow m=1,m^{\prime }=1$

$\text{Slopes of both lines are equal}$

$\therefore \text{Lines are parallel.}$

$\left(iii\right)3x+2y-4=0and6x+4y-8=0$

$y=\frac{-3}{2}x+\frac{4}{2},y=\frac{-6}{4}x+\frac{8}{4}$

$y=\frac{-3}{2}x+2,y=\frac{-3}{2}x+2$

$\Rightarrow \text{Lines are coincident}$

$\text{Because}{m}_1=m_2=\frac{-3}{2}$

$\text{Intercept = 2 in both line.}$